times as great in the second path, while notwithstanding the force remains the same as before. During the time that the first body has taken to pass through the first division, the body of fourfold mass will evidently only have passed through one-fourth of this division, and (since under a constant force the spaces described vary as the squares of the times) in double this time it will have just passed through its division. The time of passage through the first division is therefore double in the case of the body of fourfold mass. It is also apparent that while the mean velocity of passage through the first division is only one-half in the case of the body of fourfold mass, the velocity with which it enters the next division will also only be one-half of that for the other body. Thus in the case of the body of fourfold mass the velocity with which it enters one division, as well as its increment of velocity in that division, will always be only one half of the corresponding elements for the less massive body —in fine, the time of passage of the more massive body through any one of its divisions will be double of that for the other body, and hence the whole time of passage through all the divisions will likewise be double in the case of the body of fourfold mass. Thus by keeping the force the same and increasing the mass four times, the time of vibration is doubled. It is easily seen that in like manner, by keeping the mass the same and diminishing the force to one quarter of its former amount, the time of vibration will be likewise doubled. mass In fine, the time of vibration depends upon the ratio between the mass and the force, or on in such a manner that oy increasing the mass four times we double the time. Therefore the time of vibration o mass force. We may illustrate what we have now said by fastening a steel rod at one end and striking the other, when the vibrations will be very rapid; but if the end be loaded with a lump of lead the vibrations will be very slow. 133. Wave Motion.-Hitherto we have been considering the vibration of a single body or particle only; let us now consider a line of particles propagating what is termed an undulation or wave. If we cause an ordinary corkscrew to turn round upon its axis, we perceive the progress of such a form or wave from the one end to the other of the screw: nevertheless we are perfectly certain that no individual particle of the screw has travelled from the one end to the other. To use the words of a well-known writer, an undulation or wave consists in the continued transmission of a relative state of particles, while the motion of each particle separately considered is a reciprocating motion. There are many familiar examples of this kind of motion which will at once occur to our readers. Thus, for instance, when the wind blows over a field of corn, we see a form progressing across the field, but we know that notwithstanding this the indi-, vidual ears of corn do not move from their places, but only vibrate backwards and forwards. In like manner, if we throw a stone into a pool, we perceive a series of waves spreading outwards from the centre of disturbance, while at the same time a little reflection will convince us that the individual particles of water do not so move. 134. Up and Down Waves.-In order to illustrate wave motion, let us first take a case where the motion of individual particles is at right angles to the direction of transmission, as in a wave such as may be seen proceeding over the surface of a pool. Let the upper line of figure 43 represent a series of waves of this kind, the particles 1, 11, 21, &c. being at the extremity of their upward motion, at the same time that the particles 6, 16, &c. are at the extremity of their downward motion, so that I, 11, 21, &c. form the crests of the waves, and 6, 16, &c. their troughs, the distance between two contiguous crests, such as I and II, or between two contiguous hollows, such as 6 and 16, constituting what is termed a wavelength. Let us now suppose that a short time has elapsed, and that when we again view the phenomenon we find it as in the second line of Fig. 43. The particle 1 has now descended, while the particle 3 is at the summit of its upward motion, and constitutes the top of the wave. In fine, the state of things of the upper line has been pushed forward in a direction from left to right by the breadth between 1 and 3, or the wave has appeared to move over this distance. In the next figure the undulations have moved forward yet 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 FIG. 43 another step, until now the particles that were at first at the extreme limit of their downward motion have attained the extreme limit of their upward motion, and the state of things has progressed a distance equal to half a wave-length since we first viewed it. This progression of a relative state of particles will continue to go on in the same direction, from left to right, until the wave whose summit was at 1, shall have travelled over a whole wave-length, and taken up the K position II; while the wave-crest at II will have travelled to 21, and, in fact, everything will have travelled forward one complete wave-length. If we now pause to regard the state of things, we shall find that the particles have come into the very same position which they had at first, 1, 11, 21, &c., representing crests, and 6, 16, &c., representing hollows. During this interval, therefore, each particle must have performed a complete double vibration. It thus appears that, during the time which the wave has taken to travel over one complete wavelength, each particle has made a complete double vibration; so that if / denote the wave-length, or distance from one crest to the next, and v denote the velocity with which the wave is propagated, and t be the time of a complete double vibration of a particle, then 7, or one wave-length, will have been travelled over by the wave in the time t,—that is to say, 1 = v t. 135. Waves of Condensation and Rarefaction.-But we may have other waves than those now described ; for instance, the motion of a particle may not be in a direction perpen FIG. 44. dicular to that of transmission, but it may be in the same direction; the wave may, indeed, not be one of up-and-down motion, but of backward and forward motion,-in fact, a wave of condensation and rarefaction. The progress of such a wave is seen in Fig. 44, where the wave-length as well as the states of progress exhibited are analogous to those already given in the case of an up-anddown wave. Here also we see that a complete vibration of a particle is performed in the time during which the undulation progresses one wave-length. 136. Having described what is meant by wave-length, we shall now explain what is meant by the phase of a vibrating particle. The phase of a particle at any given moment denotes its place and direction of motion at that moment, as regards its vibration. If it should happen to be at its point of rest, that would be one phase; if at the extremity of its upper range, that would denote another phase; while half-way between the two would be a third, and so on. But in order to represent the phase of a vibrating particle with perfect precision, it is necessary to have recourse to a mathematical expression. It will be seen that in an undulation no two contiguous particles in the direction in which the wave is advancing are in the same phase; this, indeed, is the essential peculiarity of an undulation, for if all these particles were at precisely the same moment pulled in the same direction and to the same extent from their original positions of rest, the motion would be that of the whole body and not an undulation. It is, indeed, the distortion implied by the fact that different particles are differently placed at the same moment that gives rise to the forces which propagate the motion. When a wave is advancing,-by the front of this wave we mean all those portions of it that are in the same phase. Thus on the sea-shore the well-known ridge of water forms the front of the advancing wave. In general we may regard the line of front as perpendicular to the direction in which the wave is advancing. 137.-The amplitude means the extent of vibration of any particle on either side of its position of rest. Now, we have seen (Art. 131) that the time of vibration of the particles of elastic bodies is independent of the extent or amplitude, and we thus see that we may have two sizes of waves, in both of which the wave-length and velocity of propagation shall be the same, although the extent of vibration of the individual particles is very different for the two. If the undulation be like that of Fig. 43, we can imagine the wave-length, or distance between I and II, to remain the same, while the amount of elevation and depression of the various particles is much altered and again, if the wave be like that of Fig. 44, we can equally well imagine the wave-length to |